The present disclosure relates generally to improving battery performance by modeling the discharge behavior of a battery, and more specifically, to improving battery performance by efficiently modeling the discharge behavior of a battery using a coordinate transformation and collocation method.
Mathematical modeling and simulation of the operation of lithium ion batteries is not trivial, as concentration and potential fields must be evaluated simultaneously in both solid and liquid phases. This is complicated by the fact that the transport and kinetic parameters which determine battery behavior are highly nonlinear, leading to very complex governing equations. A general model based on concentrated solution theory to describe the internal behavior of a lithium-ion sandwich consisting of positive and negative porous electrodes, a separator, and current collectors has been developed. However, the general model is based on a number of difficult to solve equations that may take several minutes to solve.
For analysis and control of lithium-ion batteries in hybrid environments (e.g. with a fuel cell, capacitor, or other electrical components), there is a need to simulate state of charge, state of health, and other parameters of lithium-ion batteries in milliseconds. Full-order physics-based models may simulate discharge curves in several seconds to minutes, depending on the solvers, routines, computers, etc. In contrast, empirical models (based on correlations of past data) can simulate specific operating scenarios in milliseconds. However, use of these models under a different operating condition than for which they were developed may cause abuse or underutilization of electrochemical power sources, leading to significant errors.
The porous electrode model is a physics-based first principles model that describes the behavior of a 1-D battery subject to isothermal conditions. This is a system of ten partial differential equations (PDEs) in one linear coordinate, x, the radial coordinate, r, and the temporal coordinate, t, which must be solved simultaneously. The first equation is derived from concentrated solution theory and material balances. The second equation is the charge balance in the liquid phase while the third equation is the charge balance in the solid phase. The fourth equation is Fick's law of diffusion inside the solid particles (solid phase). These equations must be applied to each region of a battery cell individually, while noting that there is no active solid phase in the separator region.
In order to simplify the model, the radial dependence of the solid phase concentration can be eliminated by using a polynomial profile approximation. Rather than representing the solid phase concentration as a continuous function of x, r and t, the solid phase is represented by the particle surface concentration and the particle average concentration, both of which are functions of the linear spatial coordinate and time only. This type of volume-averaging combined with the polynomial approximation has been shown to be accurate for low to medium rates of discharge. This step eliminates some of the equations required in the model and removes the variable r from the equations, but increases the total number of equations to 12. Each of the equations that must be solved across the three regions: the positive electrode region, the negative electrode region, and the separator region. The mixed finite difference approach is known for simulation of discharge rates greater than 1 C. The mixed finite difference approach uses 6 optimally spaced node points (with 6 corresponding governing equations) to describe the behavior of the lithium ion concentration in the radial direction within the solid phase particles. This is in contrast to the polynomial profile approximation, which relies on 2 governing equations to describe the solid phase concentration. This allows the mixed finite difference approach to better capture the dynamics within the electrode at high rates, though at the cost of additional computation time. Typically, numerical approaches are used to solve these equations. The first of these solution approaches have included discretization in both space and time. Recently, discretization in space alone has been used by few researchers in order to take advantage of the speed gained by time-adaptive solvers such as DASSL for the time coordinate. This reduces the system of PDEs to a system of differential algebraic equations (DAEs) of index 1 with time as the only independent variable. However, this results in a very large number of nonlinear DAEs to be solved when a finite difference scheme is used. Assume that 50 equally spaced node points in the linear length scale (i.e. in x) are used to discretize each of the cathode, separator, and anode. The cathode now has 50 ordinary differential equations for both the electrolyte concentration and solid-phase average concentration, and 50 algebraic equations for the potential in both the electrolyte and solid phase as well as for the solid-phase surface concentration. This results in a system of 250 DAEs for the cathode. The anode is discretized in the same manner resulting in 250 additional DAEs. Since there is no active solid phase in the separator, using 50 node points will result in 50 differential equations for the electrolyte concentration and 50 algebraic equations for the electrolyte potential, for a total of 100 DAEs to describe this region. Thus, the total number of DAEs to be solved for the full-order model across the entire cell is 250+250+100=600 DAEs.
Given the large number of DAEs that must be solved, the full order spatial discretization is slow and computationally inefficient. These inefficiencies are compounded when used for control or optimization, which require fast simulation to be used effectively. Therefore it is not ideal to use a direct full order finite difference approach for these purposes. There have been many approaches to simplify the battery model for efficient evaluation while attempting to maintain a high degree of accuracy. Proper orthogonal decomposition (POD) uses the full numerical solution to fit a reduced set of eigenvalues and nodes to get a meaningful solution with a reduced number of equations. However, this method requires rigorous numerical solutions to build the POD reduced-order models. Also, once the operating condition is changed, the boundary conditions are modified, or if the parameter values are adjusted significantly, the POD model needs to be reconstructed.
In addition, a reformulated pseudo 2-D porous electrode model for galvanostatic boundary conditions has been developed. That model provided an efficient method to solve battery models in milliseconds without using a reduced order model that potentially sacrifices accuracy. This approach has proven to be useful for isothermal models, but has difficulties when non-linear properties and thermal effects are considered. The integral calculation required for Galerkin collocation becomes particularly complicated when the diffusion coefficient of the electrolyte phase is nonlinear.
Accordingly, there is a need in the art for a computationally efficient and robust method of modeling battery parameters.